Abstract
This paper deals with the question of existence for all times of the solutions of a certain class of differential equations for small initial values, and with the asymptotic behavior of these solutions. This class of equations contains different models describing the flow of viscous compressible fluids, even under the influence of a magnetic field. 1. Introduction. We consider the initial-boundary value problem on a bounded domain Ωcl with Dirichlet boundary conditions ( n) representing the relevant physical variables in their dependence on space and time. For the sake of simplicity we assume that Ωi c Rm+1 is a convex domain containing all physically reasonable values of X. The set might, e.g., include only positive values for density (which is usually the (ra +1) st component of X), and temperature. Then our equations have the form (E) X/ + /(X, VX) = LιxX + gι{x, t) (/ = 1, ... , m) ,
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